Berger, Eli

\begin{conjecture} If $M_1,\ldots,M_k$ are matroids on $E$ and $\sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1)$ for every partition $\{X_1,\ldots,X_k\}$ of $E$, then there exists $X \subseteq E$ with $|X| = \ell$ which is independent in every $M_i$. \end{conjecture}